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- W4387298322 abstract "We consider any prime number <math xmlns=http://www.w3.org/1998/Math/MathML id=M1> <mi>p</mi> </math> . Let <math xmlns=http://www.w3.org/1998/Math/MathML id=M2> <mi>k</mi> <mo>,</mo> <mi>s</mi> </math> be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference <math xmlns=http://www.w3.org/1998/Math/MathML id=M3> <mi>s</mi> </math> and length <math xmlns=http://www.w3.org/1998/Math/MathML id=M4> <mi>k</mi> </math> , where the sequence entries are from the set of quadratic residue modulo <math xmlns=http://www.w3.org/1998/Math/MathML id=M5> <mi>p</mi> </math> or the set of quadratic nonresidue modulo <math xmlns=http://www.w3.org/1998/Math/MathML id=M6> <mi>p</mi> </math> . The numbers of such sequences are denoted as <math xmlns=http://www.w3.org/1998/Math/MathML id=M7> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mfenced open=( close=) separators=|> <mrow> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M8> <msubsup> <mrow> <mi>N</mi> </mrow> <mrow> <mi>p</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msubsup> <mfenced open=( close=) separators=|> <mrow> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> </math> , respectively. In this paper, we apply analytic number theory methods, in particular, properties of Legendre’s symbol modulo <math xmlns=http://www.w3.org/1998/Math/MathML id=M9> <mi>p</mi> </math> and character sums, to study the numbers <math xmlns=http://www.w3.org/1998/Math/MathML id=M10> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mfenced open=( close=) separators=|> <mrow> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M11> <msubsup> <mrow> <mi>N</mi> </mrow> <mrow> <mi>p</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msubsup> <mfenced open=( close=) separators=|> <mrow> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> </math> . Exact formulas are given for certain values of <math xmlns=http://www.w3.org/1998/Math/MathML id=M12> <mi>k</mi> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M13> <mi>s</mi> </math> under some restrictions. In addition, estimation formulas in other cases are given." @default.
- W4387298322 created "2023-10-04" @default.
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- W4387298322 date "2023-10-03" @default.
- W4387298322 modified "2023-10-04" @default.
- W4387298322 title "The Distribution Properties of Consecutive Quadratic Residue Sequences" @default.
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- W4387298322 doi "https://doi.org/10.1155/2023/5253261" @default.
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